Volume 23, Issue 3-4

A distance-regular graph is called geometric if each its edge belongs to a unique maximal clique on the size of which the Hoffman-Delsarte inequality is satisfied as equality. A geometric classification of distance-regular graphs containing no 4-claws was put forward by S. Bang. In the present paper we refine the description of one class of such graphs. In particular, it is shown that a graph with the intersection array {45, 30, 7; 1, 2, 27} does not exist. This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 12-01-00012), the Department of Mathematical Sciences of RAS (grant no. 12-T-1-1003), the Joint Research Program of the Ural and Siberian Branches of RAS (grant no. 12-C-1-1018), the International Joint Research Fund between NSFC (China) and RFBR (grant no. 12-01-91155). The first author was supported by the Council of the President of the Russian Federation for the Support of Young Scientists (grant no. MK- 938.2011.1).

We study the dependence of complexity of the independent set problem depending on the asymptotic growth of the packing number of graphs (as a function of the number of vertices). It is shown that under some natural assumptions this problem is solvable in polynomial time if and only if the order of growth of the packing number is not than higher the logarithm of the number of vertices. This work was supported by the Russian Foundation for Basic Research, project No. 10- 01-00357-a, 11-01-00107-a, and 12-01-00749-a, the Federal Target Programme “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” for 2009-2012 (state contract No. 16.740.11.0310 and 14.B37.21.0393), and the Laboratory of Algorithms and Technologies for Networks Analysis at the National Research University “Higher School of Economics” (a grant of the Government of the Russian Federation, contract no. 11.G34.31.0057).

Let STSP m be the TSP (traveling salesman polytope) for m cities, and let CUT n be the cut polytope of the complete n-vertex graph. It is shown that CUT n is affinely equivalent to some face of the polytope STSP m with m = (2n−2)(2n−3). On the other hand, STSP m is shown to be an affine image of some face of the polytope CUT n with n = (m − 1)2 + 1.

For a constraint matrix T s,k (n) of the k-index s-ary transport problem, where each index takes n values, we consider behavior of the function α j (T s,k (n)) being equal to the mean value of square of α j th order minor of the matrix T s,k (n). Two-sided estimates of the logarithm of αj(T s,k (n)) are established. On this base, classification values j such that α j (T s,k (n))) tends to infinity and such that α j (T s,k (n)) tends to zero is obtained. An estimate of the maximal value of minors of the matrix T s,k (n) is given. Asymptotics of the function α A (T s,k (n)) being equal to the mean value of square of the minor constructed on the basis system of rows of matrix T s,k (n) is found. Main part of these results was announced earlier without proofs. The work was supported by the Russian Foundation for Basic Research, project 09-01- 00545-a.

A criterion for the finiteness of the number of states on an automaton realizing a deterministic function is established. The criterion is formulated in terms of formal power series. This work was supported by the Russian Foundation for Basic Research (project no. 12- 01-00680-a)

Two-dimensional discrete logarithm problem in a finite group G with addition as the group operation is a generalization of the classical discrete logarithm problem and consists in solving the equation Q = n 1 P 1 + n 2 P 2 with respect to n 1 , n 2 for the specified [xxx] under the assumption that there exists solution with −N 1 ≤ n 1 ≤ N 1 , −N 2 ≤ n 2 ≤ N 2 . In 2004, Gaudry and Schost have proposed an algorithm solving this problem with the average complexity [xxx] of group operations in G where (under standard heuristic assumptions) c ⋲ 2.43, N = 4N 1 N 2 , N → 1. In 2009, Galbraith and Ruprai improved this algorithm to obtain c ⋲ 2.36. In this paper we prove that for the group of points of an elliptic curve over a finite prime field having an effective automorphism φ of order 6 the average complexity of the Gaudry- Schost algorithm for the two-dimensional discrete logarithm problem with P 2 = φ (P 1 ) and N 1 = N 2 is at most [xxx], where c ⋲ 0.9781.

A class of subsets (P-sets) in the functional system of automaton functions is considered. These P-sets are defined using the Post classes. The paper is devoted to the construction of efficient completeness criteria for infinite automaton function systems M such that N ⊆ M,M \ N is finite and N is a P-set. The criterion should determine whether the system is complete via a finite set M \ N. A number of decidable cases are presented.

We describe a method of construction of circuits over some special basis admitting complete tests detecting arbitrary constant faults at the outputs of gates and having length not exceeding 4.

A system of linear equations over a ring R is called factorially solvable if for any proper ideal I of R its factorsystem is solvable over the ring R/I. A ring is called factorially solvable if any factorially solvable system over this ring is solvable. In this article it is shown that any decomposable ring is factorially solvable, a commutative principal ideal domain is factorially solvable if and only if it is subdirectly indecomposable, and that a finite commutative ring is factorially solvable if and only if it is not local.

This paper continues the study of quasigroups isotopic to groups and to Abelian groups. For identities with permutations of a certain kind the notion of the type is introduced; identities with permutations of various types are presented, which characterize quasigroups isotopic to Abelian groups. The approach being used had allowed to establish a number of new unbalanced identities involving four variables in an equasigroup, as well as a number of identities involving three variables in an equasigroup with one nullary operation, each of which characterises a quasigroup isotopic to a group or to an Abelian group.

We study quadrics and quadratic forms in a projective space over a local ring R = 2R whose maximal ideal is principal and nilpotent. The problem of enumeration of classes of projectively equivalent quadrics is completed for |R * : R *2 | = 2; when |R * : R *2 | = 4, the problem is solved together with the enumeration problem of classes of projectively congruent quadrics under an additional condition. This research was carried out with the support of the the Russian Foundation for Basic Research (project 12-01-00968) and theMinistry of Education and Science (project 1.34.11).